Googology Wiki:LoG draft
This is a draft of a rewrite to the List of googologisms page, which will probably be renamed to List of large numbers. Not much here yet. Changelog: * ??/??/15: created this page and added the structure. added uncomputable numbers. * ??/??/15: added several transfinite numbers. * ??/10/15: added proof-theoric ordinals. * 25/10/15: added most of BEAF numbers coined with LAN. added ill-defined BEAF googologisms. * 26/10/15: finished adding all BEAF googologisms by Bowers himself (Aarex / Cookiefonster / ... googologisms not added yet) * 4/2/16: E#, xE# and E^ numbers added * 4/5/16: xE^ numbers added * 4/7/16: HAN numbers added This article is a master list of all named or otherwise significant large numbers. Most googological numbers are defined by specialized notations, so here we group these numbers by notation, and the notations themselves are ordered according to their strength. It is therefore important to note that these orderings are rough approximations, and more importantly, these numbers are not necessarily in ascending order. With each number we list the author and the year the number was coined (if applicable). "n." is short for "named by," used when the number is defined by one person and named by another. Primitive recursive arithmetic These numbers are small enough to be succinctly described by primitive recursive notations. Hyper-E notation is used for some of the larger numbers here. 0 to googol * Zero, 0 * Googolminex 10-10100 (Conway and Guy, 19??) * Champernowne constant C10 = 0.1234567891011... * One, 1 * Two, 2 * Three, 3 * Four, 4 * Five, 5 * Six, 6 * Seven, 7 * Eight, 8 * Nine, 9 * Ten, 10 * Hundred, 100 * Eleventy, 110 * Twelfty or long hundred, 120 * Gross, 144 * Baker's gross, 169 * Poulter's gross, 196 (André Joyce/Michael Halm, 2004) * Short ream, 480 * Ream, 500 * Beast number, 666 * Thousand, 1000 * Great gross, 1728 = 123 * Hardy-Ramanujan Number, 1729 (????, ????) * Great Baker's gross, 2197 (Joyce, 2004) * Poulter's great gross, 2744 (Joyce, 2004) * Kaprekar's constant, 6174 * Tetrafact, 6561 = 38 * Fourth perfect number, 8128 * Myriad, 10,000 * Fugathree, 19,683 = 39 * Largest 16-bit signed integer, 32,767 * Fzsix, 46,656 = 66 * 216 = 65536 * Lakh, 100,000 * Integral-megaseptile, 142,857 * Fzseven, 823,543 * Million, 1,000,000 Googol to googolplex * Googol, 10100 (Edward Kasner & Milton Sirotta, c. 1940) * Faxul, 200! Googolplex and greater * Googolplex, 1010100 (Edward Kasner & Milton Sirotta, c. 1940) * Kilofaxul, (200!)! * Googolduplex, E100#3 (several independent inventors) * Megafaxul, ((200!)!)! * Googoltriplex, E100#4 (several independent inventors) * Gigafaxul, (((200!)!)!)! * Terafaxul, ((((200!)!)!)!)! * Petafaxul, (((((200!)!)!)!)!)! * Exafaxul, ((((((200!)!)!)!)!)!)! Hyper-E notation * Grangol, E100#100 * Tria-taxis, E1#1#3 * Grangolplex, E100#101 * Grangolgong, E100000#100000 * Googoldex, E100#1#2 * Grangoldex, E100#100#2 * Tetra-taxis, E1#1#4 * Grangoldudex, E100#100#3 * Grangoltridex, E100#100#4 * Greagol, E100#100#100 * Tria-petaxis, E1#1#1#3 * Greagolthrex, E100#100#100#2 * Greagolduthrex, E100#100#100#3 * Gigangol, E100#100#100#100 * Gigangoltetrex, E100#100#100#100#2 * Gigangoldutetrex, E100#100#100#100#3 * Gorgegol, E100#100#100#100#100 * Gulgol, E100#100#100#100#100#100 * Gaspgol, E100#100#100#100#100#100#100 * Ginorgol, E100#100#100#100#100#100#100#100 * Gargantuul, E100#100#100#100#100#100#100#100#100 * Googondol, E100#100#100#100#100#100#100#100#100#100 Hyperfactorial Array Notation * Grand Faxul, (((...(((200!)!)!)...)!)!)! w/ Faxul pairs of parentheses * Grand Kilofaxul, (((...(((200!)!)!)...)!)!)! w/ Kilofaxul pairs of parentheses * Grand Megafaxul, (((...(((200!)!)!)...)!)!)! w/ Megafaxul pairs of parentheses * Bigrand Faxul, (((...(((200!)!)!)...)!)!)! w/ Grand Faxul pairs of parentheses * Bigrand Kilofaxul, (((...(((200!)!)!)...)!)!)! w/ Grand Kilofaxul pairs of parentheses * Bigrand Megafaxul, (((...(((200!)!)!)...)!)!)! w/ Grand Megafaxul pairs of parentheses * Trigrand Faxul, (((...(((200!)!)!)...)!)!)! w/ Bigrand Faxul pairs of parentheses * Expofaxul, 200!1 * Kiloexpofaxul, (200!1)!1 * Megaexpofaxul, ((200!1)!1)!1 * Gigaexpofaxul, (((200!1)!1)!1)!1 * Teraexpofaxul, ((((200!1)!1)!1)!1)!1 * Petaexpofaxul, (((((200!1)!1)!1)!1)!1)!1 * Exaexpofaxul, ((((((200!1)!1)!1)!1)!1)!1)!1 * Grand expofaxul, (((...(((200!1)!1)!1)...)!1)!1)!1 w/ Expofaxul pairs of parentheses * Grand kiloexpofaxul, (((...(((200!1)!1)!1)...)!1)!1)!1 w/ Kiloexpofaxul pairs of parentheses * Bigrand expofaxul, (((...(((200!1)!1)!1)...)!1)!1)!1 w/ Grand expofaxul pairs of parentheses * Tetrofaxul, 200!2 * Kilotetrofaxul, (200!2)!2 * Megatetrofaxul, ((200!2)!2)!2 * Grand tetrofaxul, (((...(((200!2)!2)!2)...)!2)!2)!2 w/ Tetrofaxul pairs of parentheses * Pentofaxul, 200!3 Peano arithmetic These numbers are small enough to be succinctly described by notations provably recursive in Peano arithmetic. * Little Graham (Ronald Graham, 1971; n. Sbiis Saibian 20??) * Graham's number (Ronald Graham, 1977) Extended Hyper-E notation * Gugold, E100##100 * Gugoldagong, E100000##100000 * Great googol, E100##(E100) * Gugolda-suplex, E100##100#2 * Gugolda-dusuplex, E100##100#3 * Graatagold, E100##100#100 * Graatagolda-sudex, E100##100#100#2 * Greegold, E100##100#100#100 * Grinningold, E100##100#100#100#100 * Gugolthra, E100##100##100 * Graatagolthra, E100##100##100#100 * Greegolthra, E100##100##100#100#100 * Gugoltesla, E100##100##100##100 * Graatagoltesla, E100##100##100##100#100 * Gugolpeta, E100##100##100##100##100 * Throogol, E100###100 * Thrangol, E100###100#100 * Threagol, E100###100#100#100 * Thrugold, E100###100##100 * Thraatagold, E100###100##100#100 * Thrugolthra, E100###100##100##100 * Thrugoltesla, E100###100##100##100 * Throotrigol, E100###100###100 * Throotergol, E100###100###100###100 * Tetroogol, E100####100 * Pentoogol, E100#####100 * Hexoogol, E100######100 BEAF: Linear arrays These numbers have been coined by Jonathan Bowers in 2002~2008 using the (linear) array notation. The values added using up-arrow notation are exactly equal to the numbers. * Decker {10,10,2} = 10↑↑10 * Giggol {10,100,2} = 10↑↑100 * Tritri {3,3,3} = {3,7625597484987,2} = 3↑↑7625597484987 * Giggolplex = {10,giggol,2} = 10↑↑(10↑↑100) * Giggolduplex = {10,giggolplex,2} = 10↑↑(10↑↑(10↑↑100))) * Gaggol {10,100,3} = 10↑↑↑100 * Gaggolplex {10,gaggol,3} = 10↑↑↑(10↑↑↑100)) * Tritet {4,4,4} = 4↑↑↑↑4 = 4↑↑↑(4↑↑↑(4↑↑↑4)))) * Gaggolduplex {10,gaggolplex,3} = 10↑↑↑(10↑↑↑(10↑↑↑100))) * Geegol {10,100,4} = 10↑↑↑↑100 * Geegolplex {10,geegol,4} = 10↑↑↑↑(10↑↑↑↑100)) * Tripent {5,5,5} = 5↑↑↑↑↑5 * Gigol {10,100,5} = 10↑↑↑↑↑100 * Gigolplex {10,gigol,5} * Goggol {10,100,6} = 10↑↑↑↑↑↑100 * Goggolplex {10,goggol,6} * Trisept {7,7,7} = 7↑↑↑↑↑↑↑7 * Gagol {10,100,7} = 10↑↑↑↑↑↑↑100 * Gagolplex {10,gagol,7} * Tridecal {10,10,10} = 10↑↑↑↑↑↑↑↑↑↑10 * Boogol {10,10,100} * Boogolplex {10,10,boogol} * Boogolduplex {10,10,boogolplex} * Boogoltriplex {10,10,boogolduplex} * Corporal {10,100,1,2} * Corporalplex {10,corporal,1,2} * Grand tridecal {10,10,10,2} * Biggol {10,10,100,2} (coined by Chris Bird) * Biggolplex {10,10,biggol,2} * Biggolduplex {10,10,biggolplex,2} * Tetratri {3,3,3,3} * Baggol {10,10,100,3} * Baggolplex {10,10,baggol,3} * Beegol {10,10,100,4} * Beegolplex {10,10,beegol,4} * Bigol {10,10,100,5} * Bigolplex {10,10,bigol,5} * Boggol {10,10,100,6} * Boggolplex {10,10,boggol,6} * Bagol {10,10,100,7} * Bagolplex {10,10,bagol,7} * Supertet {4,4,4,4} * General (tetradecal) {10,10,10,10} * Generalplex {10,10,10,general} * Troogol {10,10,10,100} (coined by Chris Bird) * Troogolplex {10,10,10,troogol} * Troogolduplex {10,10,10,troogolplex} * Trigol {10,10,10,100,2} * Trigolplex {10,10,10,trigol,2} * Pentatri {3,3,3,3,3} * Traggol {10,10,10,100,3} * Traggolplex {10,10,10,traggol,3} * Treegol {10,10,10,100,4} * Treegolplex {10,10,10,treegol,4} * Superpent {5,5,5,5,5} * Trigol {10,10,10,100,5} * Trigolplex {10,10,10,trigol,5} * Troggol {10,10,10,100,6} * Troggolplex {10,10,10,troggol,6} * Tragol {10,10,10,100,7} * Tragolplex {10,10,10,tragol,7} * Pentadecal {10,10,10,10,10} * Quadroogol {10,10,10,10,100} * Quadroogolplex {10,10,10,10,quadroogol} * Quadrigol {10,10,10,10,100,2} * Hexatri {3,3,3,3,3,3} * Quadraggol {10,10,10,10,100,3} * Quadreegol {10,10,10,10,100,4} * Quadrigol {10,10,10,10,100,5} * Quadroggol {10,10,10,10,100,6} * Quadragol {10,10,10,10,100,7} * Superhex {6,6,6,6,6,6} * Hexadecal {10,10,10,10,10,10} * Quintoogol {10,10,10,10,10,100} * Quintiggol {10,10,10,10,10,100,2} * Quintaggol {10,10,10,10,10,100,3} * Quinteegol {10,10,10,10,10,100,4} * Quintigol {10,10,10,10,10,100,5} * Heptatri {3,3,3,3,3,3,3} * Supersept {7,7,7,7,7,7,7} * Heptadecal {10,10,10,10,10,10,10} * Sextoogol {10,10,10,10,10,10,100} * Superoct {8,8,8,8,8,8,8,8} * Octadecal {10,10,10,10,10,10,10,10} * Septoogol {10,10,10,10,10,10,10,100} * Superenn {9,9,9,9,9,9,9,9,9} * Ennadecal {10,10,10,10,10,10,10,10,10} * Octoogol {10,10,10,10,10,10,10,10,100} * Iteral (superdecal) {10,10,10,10,10,10,10,10,10,10} = {10,10 (1) 2} * Ultatri {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} = {3,27 (1) 2} Cascading-E notation * Godgahlah, E100#^#100 * Godgahlahgong, E100000#^#100000 * Grand godgahlah, E100#^#100#2 * Grandgahlah, E100#^#100#100 * Greagahlah, E100#^#100#100#100 * Gugoldgahlah, E100#^#100##100 * Gugolthragahlah, E100#^#100##100##100 * Throogahlah, E100#^#100###100 * Tetroogahlah, E100#^#100####100 * Gotrigahlah, E100#^#100#^#100 * Gotergahlah, E100#^#100#^#100#^#100 * Godgoldgahlah, E100#^#*#100 * Deutero-godgahlah, E100#^#*#^#100 * Gridgahlah, E100#^##100 * Gridtrigahlah, E100#^##100#^##100 * Deutero-gridgahlah, E100#^##*#^##100 * Kubikahlah, E100#^###100 * Quarticahlah, E100#^####100 * Godgathor, E100#^#^#100 * Deutero-godgathor, E100#^#^#*#^#^#100 * Godgridgathor, E100#^(#^#*##)100 * Godgathordeus, E100#^(#^#*#^#)100 * Gralgathor, E100#^#^##100 * Gralgathordeus, E100#^(#^##*#^##)100 * Thraelgathor, E100#^#^###100 * Thraelgathordeus, E100#^(#^###*#^###)100 * Terinngathor, E100#^#^####100 * Godtothol, E100#^#^#^#100 * Graltothol, E100#^#^#^##100 * Godtertol, E100#^#^#^#^#100 * Godtopol, E100#^#^#^#^#^#100 BEAF: Dimensional arrays These numbers are coined using BEAF up to multidimensional arrays, i.e. under {b,p (0,1} 2}. * Goobol {10,100 (1) 2} * Dupertri {3,tritri(1)2} = {3,3↑↑↑3 (1) 2} * Duperdecal (iteralplex) {3,iteral (1) 2} = {3,{3,27 (1) 2} (1) 2} * Goobolplex {10,goobol (1) 2} * Truperdecal {10,duperdecal (1) 2} * Quadruperdecal {10,truperdecal (1) 2} * Gibbol {10,100,2 (1) 2} * Latri {3,3,3 (1) 2} * Gabbol {10,100,3 (1) 2} * Geebol {10,100,4 (1) 2} * Gibol {10,100,5 (1) 2} * Gobbol {10,100,6 (1) 2} * Gabol {10,100,7 (1) 2} * Boobol {10,10,100 (1) 2} * Bibbol {10,10,100,2 (1) 2} * Babbol {10,10,100,3 (1) 2} * Beebol {10,10,100,4 (1) 2} * Bibol {10,10,100,5 (1) 2} * Bobbol {10,10,100,6 (1) 2} * Babol {10,10,100,7 (1) 2} * Troobol {10,10,10,100 (1) 2} * Tribbol {10,10,10,100,2 (1) 2} * Trabbol {10,10,10,100,3 (1) 2} * Treebol {10,10,10,100,4 (1) 2} * Tribol {10,10,10,100,5 (1) 2} * Trobbol {10,10,10,100,6 (1) 2} * Trabol {10,10,10,100,7 (1) 2} * Quadroobol {10,10,10,10,100 (1) 2} * Quadribbol {10,10,10,10,100,2 (1) 2} * Quadrabbol {10,10,10,10,100,3 (1) 2} * Quadreebol {10,10,10,10,100,4 (1) 2} * Quadribol {10,10,10,10,100,5 (1) 2} * Quadrobbol {10,10,10,10,100,6 (1) 2} * Quadrabol {10,10,10,10,100,7 (1) 2} * Quintoobol {10,10,10,10,10,100 (1) 2} * Quintiggol {10,10,10,10,10,100,2 (1) 2} * Quintaggol {10,10,10,10,10,100,3 (1) 2} * Quinteegol {10,10,10,10,10,100,4 (1) 2} * Quintigol {10,10,10,10,10,100,5 (1) 2} * Quintoggol {10,10,10,10,10,100,6 (1) 2} * Quintagol {10,10,10,10,10,100,7 (1) 2} * Gootrol {10,100 (1) 3} * Bootrol {10,10,100 (1) 3} * Trootrol {10,10,10,100 (1) 3} * Quadrootrol {10,10,10,10,100 (1) 3) * Gooquadrol {10,100 (1) 4) * Booquadrol {10,10,100 (1) 4} * Gitrol {10,100,2 (1) 3} * Gatrol {10,100,3 (1) 3} * Geetrol {10,100,4 (1) 3} * Gietrol {10,100,5 (1) 3} * Gotrol {10,100,6 (1) 3} * Gaitrol {10,100,7 (1) 3} * Quadreequadrol {10,10,10,10,100,4 (1) 4} * Gooquintol {10,100 (1) 5} * Emperal {10,10 (1) 10} * Gossol {10,10 (1) 100} * Emperalplex {10,10 (1) emperal} * Gossolplex {10,10 (1) gossol} * Gissol {10,10 (1) 100,2} * Gissolplex {10,10 (1) gissol,2} * Gassol {10,10 (1) 100,3} * Gassolplex {10,10 (1) gassol,3} * Geesol {10,10 (1) 100,4} * Geesolplex {10,10 (1) geesol,4} * Gussol {10,10 (1) 100,5} * Gussolplex {10,10 (1) gussol,5} * Hyperal {10,10 (1) 10,10} * Mossol {10,10 (1) 10,100} * Hyperalplex {10,10 (1) 10,hyperal} * Mossolplex {10,10 (1) 10,mossol} * Missol {10,10 (1) 10,100,2} * Massol {10,10 (1) 10,100,3} * Meesol {10,10 (1) 10,100,4} * Mussol {10,10 (1) 10,100,5} * Bossol {10,10 (1) 10,10,100} * Bissol {10,10 (1) 10,10,100,2} * Bassol {10,10 (1) 10,10,100,3} * Beesol {10,10 (1) 10,10,100,4} * Bussol {10,10 (1) 10,10,100,5} * Trossol {10,10 (1) 10,10,10,100} * Trissol {10,10 (1) 10,10,10,100,2} * Trassol {10,10 (1) 10,10,10,100,3} * Treesol {10,10 (1) 10,10,10,100,4} * Trussol {10,10 (1) 10,10,10,100,5} * Quadrossol {10,10 (1) 10,10,10,10,100} * Quintossol {10,10 (1) 10,10,10,10,10,100} * Diteral {10,10 (1)(1) 2} (formally {10,10 (1) 1 (1) 2}) * Dubol {10,100 (1)(1) 2} * Diteralplex {10,diteral (1)(1) 2} * Dutrol {10,100 (1)(1) 3} * Duquadrol {10,100 (1)(1) 4} * Admiral {10,10 (1)(1) 10} * Dossol {10,10 (1)(1) 100} * Dossolplex {10,10 (1)(1) dossol} * Dutritri {3,3,3 (1) 3,3,3 (1) 3,3,3} = {3,3 (2) 2} * Dutridecal {10,10,10 (1) 10,10,10 (1) 10,10,10} = {10,3 (2) 2} * Xappol {10,10 (2) 2} * Xappolplex {10,xappol (2) 2} * Grand xappol {10,10 (2) 3} * Dimentri {3,3,3 (1) 3,3,3 (1) 3,3,3 (2) 3,3,3 (1) 3,3,3 (1) 3,3,3 (2) 3,3,3 (1) 3,3,3 (1) 3,3,3} * Colossol {10,10 (3) 2} * Colossolplex {10,colossol (3) 2} * Terossol {10,10 (4) 2} * Terossolplex {10,terossol (4) 2} * Petossol {10,10 (5) 2} * Petossolplex {10,petossol (5) 2} * Ectossol {10,10 (6) 2} * Ectossolplex {10,ectossol (6) 2} * Zettossol {10,10 (7) 2} * Zettossolplex {10,zettossol (7) 2} * Yottossol {10,10 (8) 2} * Yottossolplex {10,yottossol (8) 2} * Xennossol {10,10 (9) 2} * Xennossolplex {10,xennossol (9) 2} * Dimendecal {10,10 (10) 2} * Gongulus {10,10 (100) 2} * Gongulusplex {10,10 (gongulus) 2} * Gongulusduplex {10,10 (gongulusplex) 2} * Gongulustriplex {10,10 (gongulusduplex) 2} * Gongulusquadraplex {10,10 (gongulustriplex) 2} BEAF: Superdimensional arrays These numbers have been coined with BEAF up to tetrational arrays. * Dulatri {3,3 (0,2) 2} * Gingulus {10,100 (0,2) 2} * Trilatri {3,3 (0,3) 2} * Gangulus {10,100 (0,3) 2} * Geengulus {10,100 (0,4) 2} * Gowngulus {10,100 (0,5) 2} * Gungulus {10,100 (0,6) 2} * Bongulus {10,100 (0,0,1) 2} * Bingulus {10,100 (0,0,2) 2} * Bangulus {10,100 (0,0,3) 2} * Beengulus {10,100 (0,0,4) 2} * Trimentri {3,3 (0,0,0,1) 2} = {3,3 ((0,1)1) 2} * Trongulus {10,100, (0,0,0,1) 2} * Quadrongulus {10,100 (0,0,0,0,1) 2} * Goplexulus {10,100 (0,0,0,...,0,0,1) 2} (100 zeroes) * Goduplexulus {10,100 ((100)1) 2} * Gotriplexulus {10,100 ((0,0,0,...,0,0,1)1) 2} HAN * Hyperfaxul, 200!1 * Kilohyperfaxul, (200!1)!1 * Megahyperfaxul, ((200!1)!1)!1 * Grand hyperfaxul, (((...(((200!1)!1)!1)...)!1)!1)!1 w/ Hyperfaxul pairs of parentheses * Grand kilohyperfaxul, (((...(((200!1)!1)!1)...)!1)!1)!1 w/ Kilohyperfaxul pairs of parentheses * Bigrand kilohyperfaxul, (((...(((200!1)!1)!1)...)!1)!1)!1 w/ Grand hyperfaxul pairs of parentheses * Trigrand kilohyperfaxul, (((...(((200!1)!1)!1)...)!1)!1)!1 w/ Bigrand hyperfaxul pairs of parentheses * Giaxul, 200!200 * Kilogiaxul, (200!200)!200 * Grand Giaxul, (((...(((200!200)!200)!200)...)!200)!200)!200 w/ Giaxul pairs of parentheses * Grand Kilogiaxul, (((...(((200!200)!200)!200)...)!200)!200)!200 w/ Kilogiaxul pairs of parentheses * Bigrand Giaxul, (((...(((200!200)!200)!200)...)!200)!200)!200 w/ Grand Giaxul pairs of parentheses * Giabixul, 200!200,200 * Kilogiabixul, (200!200,200)!200,200 * Grand Giabixul, (...(200!200,200)!...)!200,200)! w/ Giabixul pairs of parentheses * Giatrixul, 200!200,200,200 * Giaquaxul, 200!200,200,200,200 ATR0 These numbers are small enough to be succinctly described by notations provably recursive in arithmetical transfinite recursion. Extended Cascading-E notation (ε0-Γ0) * Tethrathoth, E100#^^#100 * Tethratrithoth, E100#^^#100#^^#100 * Deutero-tethrathoth, E100#^^#*#^^#100 * Tethrafact, E100(#^^#)^#100 * Grideutertethrathoth, E100(#^^#)^##100 * Tethragodgathor, E100(#^^#)^#^#100 * Tethraduliath, E100(#^^#)^(#^^#)100 * Tethradulifact, E100(#^^#)^(#^^#*#)100 * Tethrathruliath, E100(#^^#)^(#^^#*#^^#)100 * Monster-Giant, E100(#^^#)^(#^^#)^#100 * Monster Monster-Giant, E100(#^^#)^((#^^#)^#*(#^^#)^#)100 * Monster-Grid, E100(#^^#)^(#^^#)^##100 * Monster-Hecateract, E100(#^^#)^(#^^#)^#^#100 * Super Monster-Giant, E100(#^^#)^(#^^#)^(#^^#)^#100 * Double-Super Monster-Giant, E100(#^^#)^(#^^#)^(#^^#)^(#^^#)^#100 * Terrible tethrathoth, E100(#^^#)^^#100 * Terrible terrible tethrathoth, E100((#^^#)^^#)^^#100 * Tethriterator, E100#^^#>#100 * Tethriterator-dubletetrate, E100(#^^#>#)^(#^^#>#)100 * Terrible tethriterator, E100(#^^#>)^^#100 * Tethriditerator, E100#^^#>(#+#)100 * Tethritriterator, E100#^^#>(#+#+#)100 * Tethrigriditerator, E100#^^#>##100 * Tethrispatialator, E100#^^#>#^#100 * Dustaculated-tethrathoth, E100#^^#>#^^#100 * Tristaculated-tethrathoth, E100#^^#>#^^#>#^^#100 * Tethracross, E100#^^##100 * Secundo-tethrated tethracross, E100(#^^##)^^##100 * Tethritercross, E100#^^##>#100 * Dustaculated-tethracross, E100#^^##>#^^##100 * Tethracubor, E100#^^###100 * Dustaculated-tethracubor, E100#^^###>#^^###100 * Tethrateron, E100#^^####100 * Tethrapeton, E100#^^#####100 * Tethratope, E100#^^#^#100 * Tethrarxitri, E100#^^#^^#100 * Tethrarxitet, E100#^^#^^#^^#100 Larger computable numbers These numbers are too large to be described succinctly using a notation provably recursive in ATR0, but are still described by computable notations. Extended Cascading-E notation (Γ0 and beyond) * Pentacthulhum, E100#^^^#100 * Deutero-pentacthulhum, E100#^^^#*#^^^#100 * Pentacthulhufact, E100(#^^^#)^#100 * Dutetrated-pentacthulhum, E100(#^^^#)^(#^^^#)100 * Terrible pentacthulhum, E100(#^^^#)^^#100 * Dupentated-pentacthulhum, E100(#^^^#)^^(#^^^#)100 * Horrible pentacthulhum, E100(#^^^#)^^^#100 * Horrible horrible pentacthulhum, E100((#^^^#)^^^#)^^^#100 * Pentacthuliterator, E100#^^^#>#100 * Pentacthulditerator, E100#^^^#>(#+#)100 * Pentacthulgriditerator, E100#^^^#>#^#100 * Dustaculated-pentacthulhum, E100#^^^#>#^^^#100 * Pentacthulcross, E100#^^^##100 * Dustaculated-pentacthulcross, E100#^^^##>#^^^##100 * Pentacthulcubor, E100#^^^###100 * Pentacthultope, E100#^^^#^#100 * Pentacthulto-tethrathoth, E100#^^^#^^#100 * Pentacthularxitri, E100#^^^#^^^#100 * Hexacthulhum, E100#^^^^#100 * Hexacthuliterator, E100#^^^^#>#100 * Hexacthulcross, E100#^^^^##100 * Hexacthulcubor, E100#^^^^###100 * Hexacthultope, E100#^^^^#^#100 * Hexacthularxitri, E100#^^^^#^^^^#100 * Heptacthulhum, E100#^^^^^#100 * Godsgodgulus, E100#{#}#100 * Godsgodguliterator, E100#{#}#>#100 * Godsgodgulcross, E100#{#}##100 * Godsgodgulcubor, E100#{#}###100 * Godsgodgultope, E100#{#}#^#100 * Godsgodgularxitri, E100#{#}#{#}#100 * E100#{#+1}#100 * E100#{#+2}#100 * Godsgodeus, E100#{#+#}#100 * Godsgotreus, E100#{#+#+#}#100 * E100#{##}#100 * E100#{###}#100 * The centurion, E100#{#^#}#100 * Super centurion, E100#{#^^#}#100 * Ohmygosh-ohmygosh-ohmygooosh, E100#{#{#}#}#100 * Blasphemorgulus, E100{#,#,1,2}100 Uncomputable numbers These numbers require hypercomputational systems to describe succinctly. Numbers here are in increasing order (heuristically decided based on strength of functions). * \(\Sigma(1000)\) (busy beaver function, for comparison) * Fish number 4, \(\text{F}_4^{63}(3)\) ("Kyodaisuu", 2002) * \(\Xi(10^6)\) (xi function, for comparison) * Rayo's number \(\text{Rayo}(10^{100})\) (Agustín Rayo, 2007) * Fish number 7, \(\text{F}_7^{63}(10^{100})\) ("Kyodaisuu", 2013) * BIG FOOT, \(\text{FOOT}^{10}(10^{100})\) ("Wojowu" and Nathan Ho, 2014; n. Sbiis Saibian) Transfinite numbers Ordinals and cardinals are two closely related systems extending the natural numbers to infinite values. Countable ordinals Note: In this subsection, PTO stands for proof theoretic ordinal. Roughly speaking, the proof theoretic ordinal of some theory \(T\) is the smallest order type that cannot be proven to exist on a recursive well-order of the natural numbers. * \(\omega\) (Georg Cantor, 1883) * \(\omega^2\), of * \(\omega^\omega\), PTO of , RCA0 and WKL0 * \(\varepsilon_0\), PTO of PA and ACA0 (Georg Cantor, 19??) * \(\varepsilon_{\varepsilon_0}\), PTO of ACA * Cantor's ordinal \(\zeta_0\) (Georg Cantor, 19??; n. Sbiis Saibian 20??)]] * Feferman-Schütte ordinal \(\Gamma_0\), PTO of ATR0 (????, ????; n. Nik Weaver, 2005Weaver, Nik (2005), Predicativity beyond \(\Gamma_0\), ''http://arxiv.org/pdf/math/0509244v3.pdf) * Ackermann ordinal \(\vartheta(\Omega^2)\) (????, ????; . Nik Weaver, 2005) * Small Veblen ordinal \(\vartheta(\Omega^\omega)\), PTO of \(\text{ACA}_0+\Pi_2^1-\text{BI}\) (Oswald Veblen, 1908 ; n. Nik Weaver, 2005) * Large Veblen ordinal \(\vartheta(\Omega^\Omega)\) (Oswald Veblen, 1908 ; n. Nik Weaver, 2005) * Bachmann-Howard ordinal \(\vartheta(\varepsilon_{\Omega + 1})\), PTO of + axiom of infinity (????, 19??) * \(\psi(\Omega_\omega)\), PTO of \(\Pi_1^1-\text{CA}_0\) * Takeuti-Feferman-Buchholz ordinal \(\psi_0(\varepsilon_{\Omega_\omega + 1})\), PTO of (????, ????; n. David Madore, 2008) * Ψ(ψᵢ(0)), PTO of \(\Pi_1^1-\text{TR}_0\) * \(\psi_{\Omega_1}(\varepsilon_{\text{I}+1})\), PTO of KP + "there exists a recursively inaccessible ordinal" (KPI) * \(\psi_{\Omega_1}(\varepsilon_{\text{M}+1})\), PTO of KP + "there exists a recursively Mahlo ordinal" (KPM) * \(\Psi_{\Omega_1}(0,\varepsilon_{\text{K}+1})\), PTO of KP + \(\Pi_3\) reflection * Limit of Taranovsky's C function * PTO of Z2 * PTO of ZFC * Omega one chess \(\omega_1^{\mathfrak{Ch}}\) (C. D. A. Evans and Joel David Hamkins, 2013) ** Omega one chess with infinitely many pieces \(\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}\) ** Omega one of 3D chess \(\omega_1^{\mathfrak{Ch}_3}\) ** Omega one of 3D chess with infinitely many pieces \(\omega_1^ _3}\) * Admissible ordinals (????, ????), ordinals \(\alpha\) such that \(L_\alpha\) is a model of KP ** Church-Kleene ordinal \(\omega_1^{\text{CK}}\), first nonrecursive ordinal and first admissible ordinal>\(\omega\) (????, 19??) ** Recursively inaccessible ordinals (????, ????) ** Recursively Mahlo ordinals (????, ????) * Infinite time Turing machine ordinals (Joel David Hamkins and Andy Lewis, 1998) ** Supremum of all writable ordinals \(\lambda\) ** Supremum of all clockable ordinals \(\gamma\) ** Supremum of all eventually writable ordinals \(\zeta\) ** Supremum of all accidentally writable ordinals \(\Sigma\) Cardinals To represent cardinals as sets, most authors define a cardinal to be the least ordinal with that cardinality. Therefore \(\omega = \aleph_0\), \(\omega_1 = \aleph_1\), etc. and the choice of notation distinguishes cardinal arithmetic and ordinal arithmetic. All uncountable ordinals studied in the literature so far have been cardinals. * Aleph-zero, \(\aleph_0 = \omega\) * First uncountable ordinal \(\aleph_1 = \omega_1\) (sometimes \(\Omega\) is used) (Georg Cantor, 1883) * First beth number \(\beth_1\) (Equal to \(\aleph_1\) assuming the continuum hypothesis, but consistent with ZFC to be a larger uncountable cardinal.) (Georg Cantor, 1883) * The least omega fixed point \(\psi_I(0) = \sup(\Omega,\Omega_\Omega,\Omega_{\Omega_\Omega},...)\) (????, ????) * The least weakly inaccessible cardinal \(I\) (????, ????) ** The least weakly hyper-inaccessible cardinal (????, ????) * The least weakly Mahlo cardinal \(M\) (Paul Mahlo, 1911) * The least weakly compact cardinal \(K\) (????, ????) * The indescribable cardinals (????, ????) * The least rank-into-rank cardinal (????, ????) Miscellany These numbers that cannot be reasonably fit into the other categories for whatever reason. * ''q(5), Laver table pseudoinverse (Ralph Laver, 1992). Existence is an unsolved problem. Higher BEAF numbers BEAF beyond tetrational arrays is ill-defined. If it were properly defined, these numbers would probably fall under "ATR0" and "Larger computable numbers." * Triakulus {3,3,3} & 3 * Kungulus {10,100,3} & 10 * Kungulusplex {10,kungulus,3} & 10 * Quadrunculus {10,100,4} & 10 * Tridecatrix {10,10,10} & 10 * Humongulus {10,10,100} & 10 * Humongulusplex {10,10,humongulus} & 10 * Golapulus {10,100} & 10 & 10 = {10,10 (100) 2} & 10 * Golapulusplex {10,100} & 10 & 10 & 10 * Big boowa {3,3,3 / 2} * Great big boowa {3,3,4 / 2} * Grand boowa {3,3,big boowa / 2} * Super gongulus {10,10 (100) 2 / 2} * Wompogulus {10,10 (10) 2 / 100} * Guapamonga 10100 && (10100 & 10) * Guapamongaplex 10guapamonga && (10guapamonga & 10) * Big hoss {L,100}100,100 = {100,100 ////...100 /'s...//// 2} = {100,100 (1)/ 2} * Grand hoss {100,100 ////...100 /'s...//// 100} * Great big hoss {big hoss, big hoss //////...big hoss /'s...////// 2} = {big hoss,big hoss (1)/ 2} * Bukuwaha {L,100100}100,100 * Goshomity {L2,100}100,100 * Good goshomity {L2,goshomity}goshomity,goshomity * Big Bukuwaha {L2,{L,100100}100,100}100,100 * Bongo Bukuwaha {L3,{L2,{L,100100}100,100}100,100}100,100 * Quabinga Bukuwaha {L4,{L3,{L2,{L,100100}100,100}100,100}100,100}100,100 * Meameamealokkapoowa {L100,10}10,10 * Meameamealokkapoowa oompa {{L100,10}10,10&L,10}10,10 Sources